Proposition 7

For every set $\mathscr{S}$ of bounded subsets of $\mathscr{K}(X;\mathbf{C})$ that is a covering of $\mathscr{K}(X;\mathbf{C})$, the space $\mathscr{M}_{\mathscr{S}}(X;\mathbf{C})$ is Hausdorff and quasi-complete.

This results from the fact that $\mathscr{K}(X;\mathbf{C})$ is barreled (TVS, III, §4, No. 2, Cor. 4 of Th. 1).

Examples of measures. — I. Atomic measures. Let X be a locally compact space, $a$ a point of X; the mapping $f\mapsto f(a)$ of $\mathscr{K}(X;\mathbf{C})$ into $\mathbf{C}$ obviously satisfies the condition (4) with $M_K=1$ for every compact subset K of X containing $a$, hence is a measure on X, which is denoted by $\varepsilon_a$; it is called the Dirac measure at the point $a$, or the measure defined by a unit mass placed at the point $a$.

More generally, let $\alpha$ be a mapping of X into $\mathbf{C}$ such that, for every compact subset K of X,

Then, for every function $f\in\mathscr{K}(X,K;\mathbf{C})$, the sum is defined, being equal to it is clear that $\mu$ is a linear form on $\mathscr{K}(X;\mathbf{C})$ and that, for $f\in\mathscr{K}(X,K;\mathbf{C})$, in other words the condition (4) is satisfied.

A measure $\mu$ on X is said to be atomic if there exists a mapping $\alpha$ of X into $\mathbf{C}$ such that

for every compact subset K of X, and such that $\mu$ is equal to the measure defined as above. If N is the set of $x\in X$ such that $\alpha(x)\neq0$, the condition imposed on $\alpha$ implies that for every compact subset K of X, $K\cap N$ is countable. One also says that $\mu$ is defined by the masses $\alpha(x)$ placed at the points $x\in N$. If one assumes that $N\cap K$ is finite for every compact set $K\subset X$, then obviously it comes to the same to say that N is a closed and discrete subspace of X, because then every point of X has a compact neighborhood containing only a finite number of points of N, and conversely, if this is the case, then every compact subset of X can be covered by a finite number of such neighborhoods. When N is closed and discrete, every atomic measure defined by a function $\alpha$ such that $\alpha(x)=0$ on $\mathbf C N$ is called a discrete measure on X (cf. §2, No. 5).